Complex Variables This page intentionally left blank Complex Variables with an introduction to CONFORMAL MAPPING and its applications Second Edition Murray R. Spiegel, Ph.D. Former Professor and Chairman, Mathematics Department Rensselaer Polytechnic Institute, Hartford Graduate Center Seymour Lipschutz, Ph.D. Mathematics Department, Temple University John J. Schiller, Ph.D. Mathematics Department, Temple University Dennis Spellman, Ph.D. Mathematics Department, Temple University Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 1964 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permis- sion of the publisher. ISBN: 978-0-07-161570-9 MHID: 0-07-161570-9 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-161569-3, MHID: 0-07-161569-5. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please visit the Contact Us page at www.mhprofessional.com. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guara.

1. Complex Variables
This page intentionally left blank
Complex Variables
with an introduction to
CONFORMAL MAPPING and its
applications
Second Edition
Murray R. Spiegel, Ph.D.
Former Professor and Chairman, Mathematics Department
Rensselaer Polytechnic Institute, Hartford Graduate Center
Seymour Lipschutz, Ph.D.
Mathematics Department, Temple University
John J. Schiller, Ph.D.
Mathematics Department, Temple University
Dennis Spellman, Ph.D.
Mathematics Department, Temple University

2. Schaum’s Outline Series
New York Chicago San Francisco
Lisbon London Madrid Mexico City
Milan New Delhi San Juan
Seoul Singapore Sydney Toronto
Copyright © 2009, 1964 by The McGraw-Hill Companies, Inc.
All rights reserved. Except as permitted under the United States
Copyright Act of 1976, no part
of this publication may be reproduced or distributed in any form
or by any means, or stored in a database or retrieval system,
without the prior written permis-
sion of the publisher.
ISBN: 978-0-07-161570-9
MHID: 0-07-161570-9
The material in this eBook also appears in the print version of
this title: ISBN: 978-0-07-161569-3, MHID: 0-07-161569-5.
All trademarks are trademarks of their respective owners.
Rather than put a trademark symbol after every occurrence of a
trademarked name, we use names in
an editorial fashion only, and to the benefit of the trademark
owner, with no intention of infringement of the trademark.
Where such designations appear in this
book, they have been printed with initial caps.
McGraw-Hill eBooks are available at special quantity discounts

3. to use as premiums and sales promotions, or for use in corporate
training programs. To contact
a representative please visit the Contact Us page at
www.mhprofessional.com.
TERMS OF USE
This is a copyrighted work and The McGraw-Hill Companies,
Inc. (“McGraw-Hill”) and its licensors reserve all rights in and
to the work. Use of this work is
subject to these terms. Except as permitted under the Copyright
Act of 1976 and the right to store and retrieve one copy of the
work, you may not decompile,
disassemble, reverse engineer, reproduce, modify, create
derivative works based upon, transmit, distribute, disseminate,
sell, publish or sublicense the work or
any part of it without McGraw-Hill’s prior consent. You may
use the work for your own noncommercial and personal use; any
other use of the work is strictly
prohibited. Your right to use the work may be terminated if you
fail to comply with these terms.
THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND
ITS LICENSORS MAKE NO GUARANTEES OR
WARRANTIES AS TO THE ACCURACY,
ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE
OBTAINED FROM USING THE WORK, INCLUDING ANY
INFORMATION THAT CAN
BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR
OTHERWISE, AND EXPRESSLY DISCLAIM ANY
WARRANTY, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED
WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A
PARTICULAR PURPOSE.
McGraw-Hill and its licensors do not warrant or guarantee that
the functions contained in the work will meet your requirements

4. or that its operation will be
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Preface
The main purpose of this second edition is essentially the same
as the first edition with changes noted below.
Accordingly, first we quote from the preface by Murray R.
Spiegel in the first edition of this text.
“The theory of functions of a complex variable, also called for
brevity complex variables or complex
analysis, is one of the beautiful as well as useful branches of
mathematics. Although originating in an
atmosphere of mystery, suspicion and distrust, as evidenced by
the terms imaginary and complex
present in the literature, it was finally placed on a sound
foundation in the 19th century through the
efforts of Cauchy, Riemann, Weierstrass, Gauss, and other great
mathematicians.”

5. “This book is designed for use as a supplement to all current
standards texts or as a textbook for a formal
course in complex variable theory and applications. It should
also be of considerable value to those taking
courses in mathematics, physics, aerodynamics, elasticity, and
many other fields of science and
engineering.”
“Each chapter begins with a clear statement of pertinent
definitions, principles and theorems together
with illustrative and other descriptive material. This is followed
by graded sets of solved and supplementary
problems. . . .Numerous proofs of theorems and derivations of
formulas are included among the solved pro-
blems. The large number of supplementary problems with
answers serve as complete review of the material
of each chapter.”
“Topics covered include the algebra and geometry of complex
numbers, complex differential and inte-
gral calculus, infinite series including Taylor and Laurent
series, the theory of residues with applications to
the evaluation of integrals and series, and conformal mapping
with applications drawn from various fields.”
“Considerable more material has been included here than can be
covered in most first courses. This has
been done to make the book more flexible, to provide a more
useful book of reference and to stimulate
further interest in the topics.”
Some of the changes we have made to the first edition are as
follows: (a) We have expanded and cor-
rected many of the sections to make it more accessible for our
readers. (b) We have reformatted the

6. text, such as, the chapter number is now included in the label of
all sections, examples, and problems.
(c) Many results are stated formally as Propositions and
Theorems.
Finally, we wish to express our gratitude to the staff of
McGraw-Hill, particularly to Charles Wall, for
their excellent cooperation at every stage in preparing this
second edition.
SEYMOUR LIPSCHUTZ
JOHN J. SCHILLER
DENNIS SPELLMAN
Temple University
v
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Contents
CHAPTER 1 COMPLEX NUMBERS 1
1.1 The Real Number System 1.2 Graphical Representation of
Real Numbers
1.3 The Complex Number System 1.4 Fundamental Operations
with
Complex Numbers 1.5 Absolute Value 1.6 Axiomatic
Foundation of the
Complex Number System 1.7 Graphical Representation of
Complex
Numbers 1.8 Polar Form of Complex Numbers 1.9 De Moivre’s

7. Theorem
1.10 Roots of Complex Numbers 1.11 Euler’s Formula 1.12
Polynomial
Equations 1.13 The nth Roots of Unity 1.14 Vector
Interpretation of
Complex Numbers 1.15 Stereographic Projection 1.16 Dot and
Cross
Product 1.17 Complex Conjugate Coordinates 1.18 Point Sets
CHAPTER 2 FUNCTIONS, LIMITS, AND CONTINUITY 41
2.1 Variables and Functions 2.2 Single and Multiple-Valued
Functions
2.3 Inverse Functions 2.4 Transformations 2.5 Curvilinear
Coordinates
2.6 The Elementary Functions 2.7 Branch Points and Branch
Lines
2.8 Riemann Surfaces 2.9 Limits 2.10 Theorems on Limits 2.11
Infinity
2.12 Continuity 2.13 Theorems on Continuity 2.14 Uniform
Continuity
2.15 Sequences 2.16 Limit of a Sequence 2.17 Theorems on
Limits of
Sequences 2.18 Infinite Series
CHAPTER 3 COMPLEX DIFFERENTIATION AND THE
CAUCHY–RIEMANN EQUATIONS 77
3.1 Derivatives 3.2 Analytic Functions 3.3 Cauchy–Riemann
Equations
3.4 Harmonic Functions 3.5 Geometric Interpretation of the
Derivative
3.6 Differentials 3.7 Rules for Differentiation 3.8 Derivatives of
Ele-
mentary Functions 3.9 Higher Order Derivatives 3.10
L’Hospital’s Rule

8. 3.11 Singular Points 3.12 Orthogonal Families 3.13 Curves 3.14
Appli-
cations to Geometry and Mechanics 3.15 Complex Differential
Operators
3.16 Gradient, Divergence, Curl, and Laplacian
CHAPTER 4 COMPLEX INTEGRATION AND CAUCHY’S
THEOREM 111
4.1 Complex Line Integrals 4.2 Real Line Integrals 4.3
Connection Between
Real and Complex Line Integrals 4.4 Properties of Integrals 4.5
Change of
Variables 4.6 Simply and Multiply Connected Regions 4.7
Jordan Curve
Theorem 4.8 Convention Regarding Traversal of a Closed Path
4.9 Green’s
Theorem in the Plane 4.10 Complex Form of Green’s Theorem
4.11 Cauchy’s Theorem. The Cauchy–Goursat Theorem 4.12
Morera’s
Theorem 4.13 Indefinite Integrals 4.14 Integrals of Special
Functions
4.15 Some Consequences of Cauchy’s Theorem
vii
CHAPTER 5 CAUCHY’S INTEGRAL FORMULAS AND
RELATED THEOREMS 144
5.1 Cauchy’s Integral Formulas 5.2 Some Important Theorems
CHAPTER 6 INFINITE SERIES TAYLOR’S AND LAURENT’S
SERIES 169

9. 6.1 Sequences of Functions 6.2 Series of Functions 6.3 Absolute
Conver-
gence 6.4 Uniform Convergence of Sequences and Series 6.5
Power Series
6.6 Some Important Theorems 6.7 Taylor’s Theorem 6.8 Some
Special
Series 6.9 Laurent’s Theorem 6.10 Classification of
Singularities
6.11 Entire Functions 6.12 Meromorphic Functions 6.13
Lagrange’s
Expansion 6.14 Analytic Continuation
CHAPTER 7 THE RESIDUE THEOREM EVALUATION
OF INTEGRALS AND SERIES 205
7.1 Residues 7.2 Calculation of Residues 7.3 The Residue
Theorem
7.4 Evaluation of Definite Integrals 7.5 Special Theorems Used
in Evalua-
ting Integrals 7.6 The Cauchy Principal Value of Integrals 7.7
Differentiation
Under the Integral Sign. Leibnitz’s Rule 7.8 Summation of
Series
7.9 Mittag–Leffler’s Expansion Theorem 7.10 Some Special
Expansions
CHAPTER 8 CONFORMAL MAPPING 242
8.1 Transformations or Mappings 8.2 Jacobian of a
Transformation
8.3 Complex Mapping Functions 8.4 Conformal Mapping 8.5
Riemann’s
Mapping Theorem 8.6 Fixed or Invariant Points of a
Transformation
8.7 SomeGeneral Transformations 8.8 Successive
Transformations 8.9 The

10. Linear Transformation 8.10 The Bilinear or Fractional
Transformation
8.11 Mapping of a Half Plane onto a Circle 8.12 The Schwarz–
Christoffel
Transformation 8.13 Transformations of Boundaries in
Parametric Form
8.14 Some Special Mappings
CHAPTER 9 PHYSICAL APPLICATIONS OF CONFORMAL
MAPPING 280
9.1 Boundary Value Problems 9.2 Harmonic and Conjugate
Functions
9.3 Dirichlet and Neumann Problems 9.4 The Dirichlet Problem
for the
Unit Circle. Poisson’s Formula 9.5 The Dirichlet Problem for
the Half
Plane 9.6
Solution
s to Dirichlet and Neumann Problems by Conformal
Mapping Applications to Fluid Flow 9.7 Basic Assumptions 9.8
The
Complex Potential 9.9 Equipotential Lines and Streamlines 9.10
Sources
and Sinks 9.11 Some Special Flows 9.12 Flow Around Obstacles
9.13 Bernoulli’s Theorem 9.14 Theorems of Blasius
Applications to
Electrostatics 9.15 Coulomb’s Law 9.16 Electric Field Intensity.

11. Electro-
static Potential 9.17 Gauss’ Theorem 9.18 The Complex
Electrostatic
Potential 9.19 Line Charges 9.20 Conductors 9.21 Capacitance
Applica-
tions to Heat Flow 9.22 Heat Flux 9.23 The Complex
Temperature
CHAPTER 10 SPECIAL TOPICS 319
10.1 Analytic Continuation 10.2 Schwarz’s Reflection Principle
10.3 Infinite
Products 10.4 Absolute, Conditional and Uniform Convergence
of Infi-
nite Products 10.5 Some Important Theorems on Infinite
Products
10.6 Weierstrass’ Theorem for Infinite Products 10.7 Some
Special Infinite
Products 10.8 The Gamma Function 10.9 Properties of the
Gamma Function
viii Contents

12. 10.10 The Beta Function 10.11 Differential Equations 10.12